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Your question is answered on the "smooth structure" Wikipedia page. en.wikipedia.org/wiki/Differential_structure As it's an open problem, moreover of the type we likely won't resolve quickly, I'm voting to close. Please click on the "faq" and "how to ask" links above for further context.
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Ryan BudneyAug 16 '11 at 17:35

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Dear Yang, I'm afraid famous open problems are off-topic on MathOverflow. This site seems to work best for questions where you think an expert somewhere might know the answer.
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S. Carnahan♦Aug 17 '11 at 8:48

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Yang, I really don't think the second part of your question is meaningful. If one constructs a manifold homeomorphic to $CP^2$, but the construction is only of a topological manifold, then one can smooth it by declaring the homeomorphism to be a diffeomorphism. I don't think there's anything more to be said.
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Tim PerutzAug 17 '11 at 14:51

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Can you always smooth topological 4-manifold? I know that not all the topological 4 manifolds are smoothable.
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YangAug 17 '11 at 15:07

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There are plenty of non-smoothable 4-manifolds. However, one that is homeomorphic to $CP^2$ has a smooth structure: that of $CP^2$.
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Tim PerutzAug 17 '11 at 15:47

1 Answer
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This is a notorious open problem. For the moment the simplest compact four-manifold that is announced to admit (infinite number of) exotic smooth structures is $S^2\times S^2$. This result is contained here : http://arxiv.org/abs/1005.3346

I have to say that I am not at all an expert in the area
(also it seems that the above paper is not yet published). On the other hand there are several published papers showing that $CP^2\sharp 3\overline{CP^2}$ admit exotic smooth structures.

Also, it might be worth to recall that by a theorem of Yau a complex surface homeomrophic to $CP^2$ always has the standard smooth structure (in other words $CP^2$ admits a unique holomorphic structure up to bi-homolorphism). While for $S^2\times S^2$ this is still unknown (is there a surface of general type homeomorphic to $S^2\times S^2$?)

It's not clear what the second part of your question means. Do you mean, "Are there constructions of bare topological 4-manifolds that are known to be homeomorphic to $\mathbb{CP}^2$ that don't have any obvious or natural candidate for a smooth structure?"
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Robert BryantAug 16 '11 at 18:36